The theory of nonlocal continuum mechanics was formally initiated
by the papers of Kroner [1]} and
Eringen, Edelen [2]}, [3]}.
Kroner [1]} indicated the relation between
nonlocal elasticity theory of materials with long range cohesive forces.
Eringen and Edelen [3]} provided derivation of
the constitutive equations for the nonlocal elasticity.
Eringen and Kim [6]} described a relation between
non-local elasticity and lattice dynamics.
Kunin described the physical aspects of nonlocal elasticity
in the book [7]}, and studied various problems in Fourier space.
In the book [8]} Eringen considered
a unified approach to field theories for
elastic solids, viscous fluids, and
heat-conducting electromagnetic solids and fluids
that include nonlocal effects.
Rogula [9]} considered the mathematical aspects of
nonlocal elasticity models,
proposed different types of nonlocal constitutive relations
between stress and strain, and applied it to various problems
in continuum mechanics.
Non-local continuum mechanics has been treated with two different approaches [9]}, [11]}:
the gradient elasticity theory (weak non-locality) and
the integral non-local theory (strong non-locality).
In this paper we discuss the gradient models
of non-locality elasticity.
Usually two classes of gradient models are distinguished
by the different signs of the strain gradient terms
in the constitutive relations for the strain \(\varepsilon _{ij}\)  and the stress \(\sigma _{ij}\) :
\(\sigma _{ij} = \Bigl ( \lambda \varepsilon _{kk} \delta _{ij} + 2 \mu \varepsilon _{ij} \Bigr )\pm l^2 \, \Delta \, \Bigl ( \lambda \varepsilon _{kk} \delta _{ij} + 2 \mu \varepsilon _{ij} \Bigr ) ,\) 
